In Example 11.1, we considered the sequence sd[n] = αnu[n], which is the impulse respons...

In Example 11.1, we considered the sequence s_d[n] = αⁿu[n], which is the impulse response of a 1^st-order all-pole system having system function

In this problem we consider the estimation of the parameters of an all-pole model for the signal s_d[n] known only over the interval 0 ≤ n ≤ M.

(a) First, consider the estimation of a 1^st-order model by the autocorrelation method. To begin, show that the autocorrelation function of the finite-length sequence s[n] = s_d[n](u[n] − u[n −M − 1]) = αⁿ(u[n] − u[n −M − 1]) is

(b) Use the autocorrelation function determined in (a) in Eq. (11.34), and solve for the coefficient a₁ of the 1^st-order predictor.

(c) You should find that the result obtained in (b) is not the exact value (i.e., a₁ ≠ α) as obtained in Example 11.1, when the autocorrelation function was computed using the infinite sequence. Show, however, that a₁ → α for M →∞.

(d) Use the results of (a) and (b) in Eq. (11.38) to determine the minimum mean-squared prediction error for this example. Show that for M → ∞the error approaches the minimum mean-squared error found in Example 11.1 for the exact autocorrelation function.

(e) Now, consider the covariance method for estimating the correlation function. Show that for p = 1, Φ_ss [i, k] in Eq. (11.49) is given by

(f) Use the result of (e) in Eq. (11.20) to solve for the coefficient of the optimum 1st-order predictor. Compare your result to the result in (b) and to the result in Example 11.1.

(g) Use the results of (e) and (f) in Eq. (11.37) to find the minimum mean-squared prediction error. Compare your result to the result in (d) and to the result in Example 11.1.